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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Branch point structure of covering maps onto nonorientable surfaces


Author: Cloyd L. Ezell
Journal: Trans. Amer. Math. Soc. 243 (1978), 123-133
MSC: Primary 55A10
MathSciNet review: 0500900
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Abstract: Let $ f:M\, \to \,N$ be a degree n branched cover onto a compact, connected nonorientable surface with branch points $ {y_1},\,{y_2},\, \ldots ,\,{y_m}$ in N, and let the multiplicities at points in $ {f^{ - 1}}({y_i})$ be $ {\mu _{i1}},\,{\mu _{i2}},\, \ldots ,\,{\mu _{i{k_i}}}$. The branching array of f, designated by B, is the following array of numbers:

\begin{displaymath}\begin{gathered}{\mu _{11}},\,{\mu _{12}},\, \ldots ,\,{\mu _... ...m1}},\,{\mu _{m2}},\, \ldots ,\,{\mu _{m{k_m}}} \end{gathered} \end{displaymath}

We show that the numbers in the branching array must always satisfy the following conditions: (1)

$\displaystyle \sum {\{ {\mu _{ij}} \,+ \,1\vert j \,=\, 1,\,2,\, \ldots ,\,{k_i}\} \,=\, n} $

, (2) $ \sum {\{ {\mu _{ij}}\vert i\,=\, 1,\,2, \ldots ,m;j\,=\, 1,\,2, \ldots ,{k_i}\} } $ is even.

Furthermore, if B is any array of numbers satisfying these conditions, and if N is not the projective plane, then there is a branched cover onto N with B as its branching array.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0500900-0
PII: S 0002-9947(1978)0500900-0
Keywords: Branched cover, branching array, multiplicity of a branched cover, nonorientable surface
Article copyright: © Copyright 1978 American Mathematical Society